Finite Information and the Fixed Point of Mutual Renewal

Any continuum description respecting a finite information bound must be band-limited and quantised in amplitude. By Shannon--Nyquist such a continuum is information-theoretically identical to discrete bookkeeping by cells with a finite alphabet: discreteness and band-limitation are two languages for one finite-information object. Because continuum fields interact, each cell must already contain the complete data from which every field is interpolated---distinct lattices share no coupling locus, and partially overlapping lattices privilege the shared points, contradicting homogeneity. The same bound eliminates every external continuum, background, and sequencing parameter; homogeneity, extended by the identity of indiscernibles to the variables carried by the cells, eliminates internal sequencers as well. With no sequencer, the renewal generates itself: the update is its own completed quarter-turn, $V_a=e^{(\pi/2)V_a}$, with $V_a^2=-I$ and period exactly~$4$. The carrier is complex and the relay is coherent; the retained quantity per renewal is the intensity of the coherent amplitude. Mutual determination is non-commutative: the renewal operators acting on a cell pairwise fail to commute and generate a non-abelian group. The only coherent notion of state---a single configuration, not an orbit---is one self-consistent under the mutual determinations of the cells: a fixed point of the mutual renewal, the constraint picture in the quantum realisation. Mixed states reduce to this fixed point (uniform and internal-Gibbs weightings are structures on it; external temperature is excluded by closure). Redundancy is excluded by closure together with homogeneity: a decoupled sector is external, an identically coupled copy is indistinguishable and the homogeneous state lives in the symmetric quotient, a differentially coupled copy is privileged; representational redundancy is likewise excluded, so the carrier is the minimal faithful real representation of the renewal group. The structure closes with each cell both cause and effect of the others, no cell, axis, or variable privileged, nothing left over. Continuum field theory is the Shannon interpolant of that renewal. The premises are the Bekenstein bound (or its holographic strengthening), Shannon--Nyquist, and the empirical locality of interaction; no additional dynamical postulate is introduced.